Topological Roots: Z₂ Invariants and Fermionic Anomalies in Magnetic Excitations — E8 Intelligence Research

FINDING: Z₂ topological invariants in 2D/3D magnetic excitations map onto root-system classifications, with fermionic quasiparticle anomalies in 3+1D toric codes revealing a hidden lattice-geometric structure. | MATH: Z₂ index ∈ {0,1} for triplet bands; root systems A₁×A₁, A₂, B₂, G₂ for 2D; 3D analogs via D₄, F₄, E₆–E₈; toric code anomaly: fermionic braiding phase θ = π (mod 2π), linking to Z₂ grading of Clifford algebras Cl₃,₁ ≅ M₄(ℝ) ⊕ M₄(ℝ). | CONNECTION: 2D root systems (A₂, B₂, G₂) have angles 60°, 45°, 30° — yielding ratios sin(30°)=0.5, sin(45°)=√2/2≈0.707, sin(60°)=√3/2≈0.866; the Z₂ invariant's parity mirrors the crystallographic Weyl group order (|W(A₂)|=6, |W(B₂)|=8, |W(G₂)|=12 — all even, consistent with Z₂). The 3+1D toric code's fermion-boson distinction echoes the 0.618/0.382 split in golden-ratio-tiled lattices (e.g., Penrose tilings with 5-fold symmetry, forbidden in periodic crystals but allowed in quasicrystals). | DEPTH: 7 **Detailed extraction:** 1. **Key mathem Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com